Preheat the oven to 350 and butter a 9×13-inch baking pan with 2-inch-high sides. Combine the first 3 ingredients in a bowl. Stir the butter and chocolate in a medium sized bowl over a saucepan of simmering water. Stir the chocolate and butter in this double boiler until melted and smooth.
Beat the eggs, sugar and vanilla in large bowl and stir in the warm chocolate mix, then add the dry ingredients. Fold in graham crackers and pour the batter into the pan. Place the 12 large marshmallows into the batter and bake until a toothpick comes out with moist crumbs attached, about 30 to 35 minutes.
Marshmallows will be browned and puffy but will deflate as the brownies cool. Cool for at least 20 minutes then slice with a knife, cleaning the knife with hot water if it gets too messy and sticky.